[alogo] 1. Generalization of Thales Theorem

Let (Y) be a bundle of circles intersecting at two points G, F. Draw two lines through G and consider the segments cut off by the circles. Then WN/NR = UV/VP = QO/OS = KJ/LK. In particular the ratio UV/VP is independent of the location of the line GE. It is also UV/WN = VP/NR. Thus, the ratio UV/WN is independent of the pair of circles through which it is defined and depends only on the position of the two lines GE and GH.
Thales theorem can be considered as a special (limit) case of this one, when the point G goes to a definite direction L to infinity and the circle bundle becomes the bundle of lines through point F. Then lines GE, GH are lines parallel to L.

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The figure speaks from itself. The four equal ratios indicate that the seemingly variable UV/VP is equal to the JK/KL which is independent of the position of the line FE. For the last two ratios note that UV/WN = (UV/JK)/(WN/JK), which is a ratio of cosines of the angles of lines UV, WN with line IJ. A ratio which is independent of the particular pair of circles defining the segments UV and WN.

[alogo] 2. Inverse

Like Thales theorem, this generalization has also an inverse:
If the ratios UV/VP = WN/NR taken on two intersecting lines GE, GH, then the three circumcircles of triangles UWF, VNF, PRF pass through a common point F.

This is easily proved by assuming that the two first circumcircles pass through G and showing that the third must also pass through G.

The inverse of the theorem has also a somewhat curious consequence. If we consider two intersecting lines GE and GH and take correspondingly equal segments starting from two arbitrary points U, W on them:
UV = VP = PM = ... = a, and
WN = NR = RT = ... = b,
then the circumcircles of the triangles GUW, GVN, GPR, GMT, ... pass all through a common point F. Besides, if we join corresponding points with lines UW, VN, PR, MT, ... we see that these lines envelope a curve resembling a parabola.

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[alogo] 3. The hidden parabola

This parabola has its focus at F and is studied in ThalesParabola.html . The previous way to construct a rough image of a parabola, by taking repeatedly segments on two lines, equal to a on one and equal to b on the other line, is encountered often in books on popularized Mathematics.
The shape of the parabola depends from the two lines GE, GH, the initial points W,U and the constants a,b. As expected from the property in (1) the dependence of the parabola on a,b is not absolute but relative. In fact, it depends only on the ratio a/b.
Another remark concerns triangles GWU, GNV, ... etc.. If we show that all these lines are tangent to a parabola, then these triangles have all their sides tangent to this parabola. But for parabolas we know that (see ParabolaSkew.html ) all triangles with sides tangent to a parabola have circumcircles passing through the focus.
This fact gives another aspect to the property connecting it with triangles tangent to a certain parabola. The property of constancy of the ratio UV/VP = WN/NR is proved to be a consequence of a more general property of conics related to the cross-ratio of four points on them.
Thus, we can say that Thales theorem is a limiting case of the present property, which in turn results from the possibility to define the cross-ratio of four points on a conic and measure it in a multitude of ways by projecting the points onto a line (see ParabolaProperty.html and the references given there).

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